DOCTORAL SEMINAR, SPRING SEMESTER 2007 Experimental Design & Analysis Further Within Designs; Mixed Designs; Response Latencies April 3, 2007

Outline Mixed, or split-plot, designs Response latency designs, a special case for split plots

Mixed Designs The mixed factorial design is, in fact, a combination of a within-subjects design and a fully-crossed factorial design  A special type of mixed design, that is particularly common, is the pre-post-control design All subjects are given a pre-test and a post-test, and these two together serve as a within-subjects factor Participants are also divided into two groups  One group is the focus of the experiment (i.e., experimental group)  Other group is a base line (i.e., control) group.

Mixed Designs Also known as split-plot designs, mixed designs originated in agricultural research  Seeds were assigned to different plots of land, each receiving a different treatment Subjects (ex., seeds) are randomly assigned to each level (ex., plots) of the between-groups factor (soil types), prior to receiving the within-subjects repeated factor treatments (ex., applications of different types of fertilizer)

Mixed Designs Partitioning the variance  Estimate between-groups effect  Estimate within-groups effect  Estimate interaction Using error terms to estimate effects, significance  Between-subjects error term is used for between groups-effect  Within-subjects error terms is used for within-groups effect  Within-subjects error terms is used for the interaction since this includes a within-subjects effect

Mixed Designs Consider mixed design in which factor A is between-subject factor and B is within-subject factor: Ax(BxS)  Subjects factor is crossed with B but nested in A  Half of subjects see a 1 b 1, a 1 b 2 and half of subjects see a 2 b 1, a 2 b 2 conditions  Sources of variability: A, B, AxB, S(A), BxS(A)  To test effects: compare mean square of effects (A, B, AxB) with mean square for effects with subjects (MS S(A), MS BxS(A), MS BxS(A) )

Mixed Designs Take Keppel’s example of kangaroo rats on p  Between-subjects: 3 kinds of rats (A)  Within-subjects: number of landmarks (B)  Each subject is tested at 4 levels of B

Mixed Designs Another way to analyze the data is to look at multivariate analysis  Treats each rat’s scores as a vector  All the b scores for a rat represents a single multivariate observation Assumptions for data  Sphericity: homogeneity of variances for within- subject data  Homogeneity of covariance: for within- and between- subject data

Mixed Designs We examine the effects of a new type of cognitive therapy on depression  Give a depression pre-test to a group of persons diagnosed as clinically depressed and randomly assign them into two groups (traditional and cognitive therapy)  After the patients were treated according to their assigned condition for some period of time, they would be given a measure of depression again (post-test)  This design consists of one within-subject variable (test), with two levels (pre- and post-), and one between-subjects variable (therapy), with two levels (traditional and cognitive)

Within-Subject Designs: Usefulness Researchers using the pre- post-control design look for an interaction such that one cell in particular stands out, and that is the experimental group’s post test score. Ideally the pre-test scores will be equivalent. It is the post-test score difference between the experimental and control that is important.

Pre-test Post-test: Alternatives One-way ANOVA on the posttest scores  Ignores pretest data and is not recommended Split-plot repeated measures ANOVA  Between-subjects factor is the group (treatment or control) and the repeated measure is the test scores for two trials. The resulting ANOVA table will include a main treatment effect (reflecting being in the control or treatment group) and a group-by-trials interaction effect (reflecting treatment effect on posttest scores, taking pretest scores into account) One-way ANOVA on difference scores  Difference is the posttest score minus the pretest score and is equivalent to a split-plot design if there is close to a perfect linear relation between the pretest and posttest scores in all treatment and control groups  This linearity will be reflected in a pooled within-groups regression coefficient of 1.0. When this coefficient approaches 1.0, this method is more powerful than the ANCOVA method. ANCOVA on the posttest scores  Using the pretest scores as a covariate control. When pooled within-groups regression coefficient is less than 1.0, the error term is smaller in this method than in ANOVA on difference scores, and the ANCOVA method is more powerful.